Mixing
Approach to prove vector spaces and subspaces of vector spaces
$begingroup$
Let $(E,+,.)$ be a $K$-vector space and $F$ a subspace of $E$, it follows that both meet certain requirements to be what they are.
My question : what is your approach in proving both to be what they are? With which condition do you start first? $0_E in F$ for a subspace as a first step perhaps? Or do you look for certain characteristics?
Generally are there indications that one rule won't be met and thus one would avoid wasting time by going through each?
abstract-algebra vector-spaces
asked Feb 2 at 17:55
LuywLuyw
365
$endgroup$
add a comment |
$begingroup$
Let $(E,+,.)$ be a $K$-vector space and $F$ a subspace of $E$, it follows that both meet certain requirements to be what they are.
My question : what is your approach in proving both to be what they are? With which condition do you start first? $0_E in F$ for a subspace as a first step perhaps? Or do you look for certain characteristics?
Generally are there indications that one rule won't be met and thus one would avoid wasting time by going through each?
abstract-algebra vector-spaces
asked Feb 2 at 17:55
LuywLuyw
365
$endgroup$
$begingroup$
It really depends on the problem you're looking at. If you are asked to prove that E is a subspace of F then it doesn't matter much where you start, since all of them will turn out to be true. So in this case I would start with the "easiest" and work your way up. If you think that one of them may fail, then try to build some intuition about why F is not a subspace and work to find the failing condition from this intuition.
$endgroup$
– Nao
Feb 2 at 18:05
add a comment |
$begingroup$
Let $(E,+,.)$ be a $K$-vector space and $F$ a subspace of $E$, it follows that both meet certain requirements to be what they are.
My question : what is your approach in proving both to be what they are? With which condition do you start first? $0_E in F$ for a subspace as a first step perhaps? Or do you look for certain characteristics?
Generally are there indications that one rule won't be met and thus one would avoid wasting time by going through each?
abstract-algebra vector-spaces
asked Feb 2 at 17:55
LuywLuyw
365
$endgroup$
Let $(E,+,.)$ be a $K$-vector space and $F$ a subspace of $E$, it follows that both meet certain requirements to be what they are.
My question : what is your approach in proving both to be what they are? With which condition do you start first? $0_E in F$ for a subspace as a first step perhaps? Or do you look for certain characteristics?
Generally are there indications that one rule won't be met and thus one would avoid wasting time by going through each?
abstract-algebra vector-spaces
abstract-algebra vector-spaces
asked Feb 2 at 17:55
LuywLuyw
365
asked Feb 2 at 17:55
LuywLuyw
365
asked Feb 2 at 17:55
LuywLuyw
365
asked Feb 2 at 17:55
LuywLuyw
365
asked Feb 2 at 17:55
LuywLuyw
365
365
$begingroup$
It really depends on the problem you're looking at. If you are asked to prove that E is a subspace of F then it doesn't matter much where you start, since all of them will turn out to be true. So in this case I would start with the "easiest" and work your way up. If you think that one of them may fail, then try to build some intuition about why F is not a subspace and work to find the failing condition from this intuition.
$endgroup$
– Nao
Feb 2 at 18:05
add a comment |
$begingroup$
It really depends on the problem you're looking at. If you are asked to prove that E is a subspace of F then it doesn't matter much where you start, since all of them will turn out to be true. So in this case I would start with the "easiest" and work your way up. If you think that one of them may fail, then try to build some intuition about why F is not a subspace and work to find the failing condition from this intuition.
$endgroup$
– Nao
Feb 2 at 18:05
$begingroup$
It really depends on the problem you're looking at. If you are asked to prove that E is a subspace of F then it doesn't matter much where you start, since all of them will turn out to be true. So in this case I would start with the "easiest" and work your way up. If you think that one of them may fail, then try to build some intuition about why F is not a subspace and work to find the failing condition from this intuition.
$endgroup$
– Nao
Feb 2 at 18:05
$begingroup$
It really depends on the problem you're looking at. If you are asked to prove that E is a subspace of F then it doesn't matter much where you start, since all of them will turn out to be true. So in this case I would start with the "easiest" and work your way up. If you think that one of them may fail, then try to build some intuition about why F is not a subspace and work to find the failing condition from this intuition.
$endgroup$
– Nao
Feb 2 at 18:05
add a comment |
1 Answer
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$begingroup$
In the case of vector spaces, the first thing that I would check is whether it is closed for addition. In the case of fields, the first things that I would check is whether the multiplication is commutative and whether each non-zero element has an inverse.
answered Feb 2 at 18:03
José Carlos SantosJosé Carlos Santos
174k23134243
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add a comment |
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$begingroup$
In the case of vector spaces, the first thing that I would check is whether it is closed for addition. In the case of fields, the first things that I would check is whether the multiplication is commutative and whether each non-zero element has an inverse.
answered Feb 2 at 18:03
José Carlos SantosJosé Carlos Santos
174k23134243
$endgroup$
add a comment |
$begingroup$
In the case of vector spaces, the first thing that I would check is whether it is closed for addition. In the case of fields, the first things that I would check is whether the multiplication is commutative and whether each non-zero element has an inverse.
answered Feb 2 at 18:03
José Carlos SantosJosé Carlos Santos
174k23134243
$endgroup$
add a comment |
$begingroup$
In the case of vector spaces, the first thing that I would check is whether it is closed for addition. In the case of fields, the first things that I would check is whether the multiplication is commutative and whether each non-zero element has an inverse.
answered Feb 2 at 18:03
José Carlos SantosJosé Carlos Santos
174k23134243
$endgroup$
In the case of vector spaces, the first thing that I would check is whether it is closed for addition. In the case of fields, the first things that I would check is whether the multiplication is commutative and whether each non-zero element has an inverse.
answered Feb 2 at 18:03
José Carlos SantosJosé Carlos Santos
174k23134243
answered Feb 2 at 18:03
José Carlos SantosJosé Carlos Santos
174k23134243
answered Feb 2 at 18:03
José Carlos SantosJosé Carlos Santos
174k23134243
answered Feb 2 at 18:03
José Carlos SantosJosé Carlos Santos
174k23134243
174k23134243
add a comment |
add a comment |
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$begingroup$
It really depends on the problem you're looking at. If you are asked to prove that E is a subspace of F then it doesn't matter much where you start, since all of them will turn out to be true. So in this case I would start with the "easiest" and work your way up. If you think that one of them may fail, then try to build some intuition about why F is not a subspace and work to find the failing condition from this intuition.
$endgroup$
– Nao
Feb 2 at 18:05